Optimal. Leaf size=449 \[ \frac {3 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {3 c \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {a^2 x^2+1}}+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}-\frac {3 c \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {9 a c x^2 \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {a^2 x^2+1}}-\frac {27 c \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {a^2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.56, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5684, 5682, 5675, 5663, 5779, 3312, 3307, 2180, 2204, 2205, 5717, 5699} \[ \frac {3 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {a^2 x^2+1}}+\frac {3 \sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {a^2 x^2+1}}+\frac {3 c \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {a^2 x^2+1}}+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}-\frac {3 c \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{32 a}-\frac {9 a c x^2 \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {a^2 x^2+1}}-\frac {27 c \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {a^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5663
Rule 5675
Rule 5682
Rule 5684
Rule 5699
Rule 5717
Rule 5779
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2} \, dx &=\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2} \, dx-\frac {\left (3 a c \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \sqrt {\sinh ^{-1}(a x)} \, dx}{8 \sqrt {1+a^2 x^2}}\\ &=-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \int \frac {\left (1+a^2 x^2\right )^{3/2}}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{64 \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \int \frac {\sinh ^{-1}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {1+a^2 x^2}}-\frac {\left (9 a c \sqrt {c+a^2 c x^2}\right ) \int x \sqrt {\sinh ^{-1}(a x)} \, dx}{16 \sqrt {1+a^2 x^2}}\\ &=-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh ^4(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (9 a^2 c \sqrt {c+a^2 c x^2}\right ) \int \frac {x^2}{\sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \, dx}{64 \sqrt {1+a^2 x^2}}\\ &=-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}\\ &=\frac {9 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a \sqrt {1+a^2 x^2}}-\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}\\ &=-\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a \sqrt {1+a^2 x^2}}\\ &=-\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{512 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{512 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}\\ &=-\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}+\frac {\left (9 c \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}\\ &=-\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.32, size = 186, normalized size = 0.41 \[ \frac {c \sqrt {a^2 c x^2+c} \left (60 \sqrt {2 \pi } \sqrt {\sinh ^{-1}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )+60 \sqrt {2 \pi } \sqrt {\sinh ^{-1}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )+384 \sinh ^{-1}(a x)^3+640 \sinh \left (2 \sinh ^{-1}(a x)\right ) \sinh ^{-1}(a x)^2-480 \sinh ^{-1}(a x) \cosh \left (2 \sinh ^{-1}(a x)\right )-5 \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},4 \sinh ^{-1}(a x)\right )+5 \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-4 \sinh ^{-1}(a x)\right )\right )}{2560 a \sqrt {a^2 x^2+1} \sqrt {\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsinh \left (a x \right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {asinh}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________